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%I #95 Jan 02 2023 05:09:23
%S 1,4,36,504,9576,229824,6664896,226606464,8837652096,388856692224,
%T 19053977918976,1028914807624704,60705973649857536,
%U 3885182313590882304,268077579637770878976,19837740893195045044224,1567181530562408558493696,131643248567242318913470464
%N Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).
%H Vincenzo Librandi, <a href="/A008546/b008546.txt">Table of n, a(n) for n = 0..300</a>
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H Keiichi Shigechi, <a href="https://arxiv.org/abs/2212.14666">On the lattice of weighted partitions</a>, arXiv:2212.14666 [math.CO], 2022. See p. 27.
%F a(n) = 4*A034301(n) = (5*n - 1)(!^5), n >= 1, with a(0) = 1.
%F a(n) = A011801(n + 1, 1) (first column of triangle).
%F a(n) ~ (sqrt(2*Pi)/Gamma(4/5))*n^(n + 3/10)*(5/e)^n*(1 + 1/(300*n) + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 9*x/(1 - 10*x/(1 - 14*x/(1 - 15*x/(1 - 19*x/(1 - 20*x/(1 - 24*x/(1 - ... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F a(n) = (-1)^n*Sum_{k = 0..n} 5^k*s(n + 1, n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012
%F G.f.: ( 1 - 1/Q(0) )/x where Q(k) = 1 - x*(5*k - 1)/(1 - x*(5*k + 5)/Q(k + 1) ); (continued fraction); e.g.f. (1 - 5*x)^(-4/5). - _Sergei N. Gladkovskii_, Mar 20 2013
%F G.f.: 1/x - G(0)/(2*x), where G(k) = 1 + 1/(1 - x*(5*k - 1)/(x*(5*k - 1) + 1/G(k + 1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013
%F a(n) = 5^n * Gamma(n + 4/5) / Gamma(4/5). - _Vaclav Kotesovec_, Jan 28 2015
%F a(n) + (-5*n + 1)*a(n - 1) = 0. - _R. J. Mathar_, Sep 04 2016
%F G.f.: 1/(1 - 4*x - 20*x^2/(1 - 14*x - 90*x^2/(1 - 24*x - 210*x^2/(1 - 34*x - 380*x^2/(1 - 44*x - 600*x^2/(1 - 54*x - 870*x^2/(1 - ...))))))) (Jacobi continued fraction). - _Nikolaos Pantelidis_, Feb 29 2020
%F Sum_{n>=0} 1/a(n) = 1 + (e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - _Amiram Eldar_, Dec 19 2022
%p f:= n-> product(5*k+4, k=0..n-1);
%t FoldList[Times, 1, 5Range[0, 20] + 4] (* _Vincenzo Librandi_, Jun 10 2013 *)
%t CoefficientList[Series[(1 - 5x)^(-4/5), {x, 0, 20}], x] Range[0, 20]! (* _Vaclav Kotesovec_, Jan 28 2015 *)
%t Table[5^n Pochhammer[4/5, n], {n, 0, 20}] (* _G. C. Greubel_, Aug 20 2019 *)
%o (PARI) vector(20, n, n--; prod(j=0,n-1, 5*j+4) ) \\ _G. C. Greubel_, Aug 20 2019
%o (Magma) [1] cat [(&*[5*k+4: k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 20 2019
%o (Sage) [5^n*rising_factorial(4/5, n) for n in (0..20)] # _G. C. Greubel_, Aug 20 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> 5*k+4 )); # _G. C. Greubel_, Aug 20 2019
%Y Cf. A011801, A034301, A048994.
%Y Cf. A008548, A047055, A047056, A051150, A052562, A254287.
%K nonn
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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